I'm trying to figure out how to limit the items I want to express to a certain number.
For example: For all apples out there, there are exactly 3 apples that are orange.
A(x) = x is an apple
O(x) = x is orange
I really have no idea how to start this, not even sure which quantifier to use. Would really appreciate some advice in how to break down these problems in an efficient way.
Thanks in advance!
Ok, several ways to do this:
Probably the most straightforward way is to say that there are three distinct orange apples and no more:
$\exists x \exists y \exists z (A(x) \land O(x) \land A(y) \land O(y) \land A(z) \land O(z) \land x \neq y \land x \neq z \land y \neq z\land \neg \exists w (A(w) \land O(w) \land w \neq x \land w \neq y \land w \neq z))$
But another way of thinking about this is: there are three distinct apples, and any object is an orange apple if and only if it is one of those:
$\exists x \exists y \exists z (x \neq y \land x \neq z \land y \neq z \land \forall w ((A(w) \land O(w)) \leftrightarrow (w = x \lor w = y \lor w = z))$
This is more efficient ... but probably a little harder to grasp.